1. ## integration of application

hey guys, im struggling with these 2 questions, think im getting brain freeze, must be missing something obvious. can any one help?

1.the charge Q COloumbs on a capacitor varies with time according to the formula

dQ/dt = -250sin100t

find the charge after 0.0005 seconds if Q=0.015 coloumbs at time t=0 seconds?

2. dQ/dt = 100cos200t

find the charge after 0.0005 seconds if Q=1 coloumbs at time t=pie/400 seconds?

and also, i know this is prob a stupid question but can someone tell me what

exp 1 =?
exp 0 =?
exp -1=?

thanks!!!

2. 1. $\displaystyle \frac{dQ}{dt} = -250\sin(100t)$ is a separable differential equation, which you can solve to get a formula for $\displaystyle Q(t)$.

Q = $\displaystyle \int dQ = \int -250\sin(100t) dt = \frac{250}{100}\cos(100t) + C= 2.5 \cos(100t) + C$.

where $\displaystyle C$ is some constant. To find this constant use the initial values given.

$\displaystyle 0.015 = Q(0) = 2.5 \cos(0) + C \Rightarrow C = 0.015 - 2.5 = -2.485$

So the equation for $\displaystyle Q(t) = 2.5 \cos(100t) -2.485$. use this to find the charge at any time. Problem 2 is basically the same problem with different values.
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$\displaystyle e^0 = 1$ since any non-zero number to the 0 is 1.
$\displaystyle e^1 = e$ since any number to the first power is itself.
$\displaystyle e^{-1} = \frac{1}{e} \approx 0.367879441$.